A330510: Triangle read by rows: T(n,k) is the number of ternary strings of length n+1 with k+1 indispensable digits and a nonzero leading digit, with 0 <= k <= n.

A330510 ↗ palindromicγ-positivenonnegative coefficients

Proved for all n all roots real, in (−∞, 0)coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence from A330510

\(P_{n}(x) = \left(2 + 2\,x\right)\,P_{n-1}(x) + \left(-1 - x - x^{2}\right)\,P_{n-2}(x)\)

\(P_{0}(x) = 2\)

\(P_{1}(x) = 3 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	2								2
1	3	3							6
2	4	10	4						18
3	5	22	22	5					54
4	6	40	70	40	6				162
5	7	65	171	171	65	7			486
6	8	98	356	534	356	98	8		1458
7	9	140	665	1373	1373	665	140	9	4374

All 15 rows b-files: A008776 (row sums), A330510, A330510 (reversed triangle) ?

Technical details

Node id: n-539ff84504d2bf3a · revision 0 · JSON
Definition source: oeis_catalog:A330510
Fingerprint: f6-e5056e0add0ce0ee

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
Palindromic rows: yes
γ-positive (gamma_positive): yes
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

A330510 triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?
A330510 reversed triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?

Row sums ? A027334, A025192, A008776, A099856, A110593

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:  2;
  1:  3,   3;
  2:  4,  10,    4;
  3:  5,  22,   22,    5;
  4:  6,  40,   70,   40,     6;
  5:  7,  65,  171,  171,    65,     7;
  6:  8,  98,  356,  534,   356,    98,    8;
  7:  9, 140,  665, 1373,  1373,   665,  140,    9;
  8: 10, 192, 1148, 3088,  4246,  3088, 1148,  192,  10;
  9: 11, 255, 1866, 6294, 11257, 11257, 6294, 1866, 255, 11;

%F Formula

The row polynomials satisfy P(n,x) = (2+2*x)*P(n-1,x) - (1+x+x^2)*P(n-2,x) for n >= 2, with P(0,x) = 2, P(1,x) = 3 + 3*x.

%t Mathematica

P[0, x_] := 2;
P[1, x_] := 3 + 3*x;
P[n_, x_] := P[n, x] = Expand[(2 + 2*x)*P[n - 1, x] - (1 + x + x^2)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 3 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 2*x)*v[i - 1] - (1 + x + x^2)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

A330510 triangle, offset 0, from P₀

%e Example

Triangle begins:
  0:  2;
  1:  3,   3;
  2:  4,  10,    4;
  3:  5,  22,   22,    5;
  4:  6,  40,   70,   40,     6;
  5:  7,  65,  171,  171,    65,     7;
  6:  8,  98,  356,  534,   356,    98,    8;
  7:  9, 140,  665, 1373,  1373,   665,  140,    9;
  8: 10, 192, 1148, 3088,  4246,  3088, 1148,  192,  10;
  9: 11, 255, 1866, 6294, 11257, 11257, 6294, 1866, 255, 11;

%F Formula

The row polynomials satisfy P(n,x) = (2+2*x)*P(n-1,x) - (1+x+x^2)*P(n-2,x) for n >= 2, with P(0,x) = 2, P(1,x) = 3 + 3*x.

%t Mathematica

P[0, x_] := 2;
P[1, x_] := 3 + 3*x;
P[n_, x_] := P[n, x] = Expand[(2 + 2*x)*P[n - 1, x] - (1 + x + x^2)*P[n - 2, x]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 3 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 2*x)*v[i - 1] - (1 + x + x^2)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

A330510 reversed triangle, offset 0, from P₀

%e Example

Triangle begins:
  0:  2;
  1:  3,   3;
  2:  4,  10,    4;
  3:  5,  22,   22,    5;
  4:  6,  40,   70,   40,     6;
  5:  7,  65,  171,  171,    65,     7;
  6:  8,  98,  356,  534,   356,    98,    8;
  7:  9, 140,  665, 1373,  1373,   665,  140,    9;
  8: 10, 192, 1148, 3088,  4246,  3088, 1148,  192,  10;
  9: 11, 255, 1866, 6294, 11257, 11257, 6294, 1866, 255, 11;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (2+2*x)*P(n-1,x) - (1+x+x^2)*P(n-2,x) for n >= 2, with P(0,x) = 2, P(1,x) = 3 + 3*x.

%t Mathematica

P[0, x_] := 2;
P[1, x_] := 3 + 3*x;
P[n_, x_] := P[n, x] = Expand[(2 + 2*x)*P[n - 1, x] - (1 + x + x^2)*P[n - 2, x]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 3 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 2*x)*v[i - 1] - (1 + x + x^2)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0)
by Liu–Wang theorem for three-term recurrences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Liu–Wang theorem for three-term recurrences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Liu–Wang theorem for three-term recurrences
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 1
by Liu–Wang theorem for three-term recurrences

Proof chains ? JSON

Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

References

[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003